Introduction

Spectral methods have been widely used in geophysical modelling of different fluids including the atmosphere, ocean, outer core and mantle. Variables are expanded as a sum of orthogonal global basis functions, typically trigonometric or polynomial, in contrast to finite difference, finite element and finite volume methods in which the basis functions are local. The convergence of the method is faster than spatial methods, meaning that high mathematical accuracy can be obtained with relatively few basis functions when representing smoothly varying fields, although sharp gradients or discontinuities can cause problems. Spherical geometry is easily treated by using spherical harmonics as basis functions, giving approximately uniform resolution over the sphere.

Basis functions are typically different in the horizontal (azimuthal) and vertical (radial) directions because of the differing boundary conditions (side boundaries are often periodic). In the vertical (radial) direction, Chebyshev polynomials or finite differences are typically used. Once expanded in harmonics, spatial derivatives are given by exact analytic expressions. For linear equations, the equations for different harmonics decouple in spectral space, and each mode can be solved independently. The method is thus ideally suited for equations in which the coefficients in front of dependent variables (e.g. viscosity, thermal diffusivity, wave velocity) are spatially constant.